Therefore, \(\lbrace  x_{S, \alpha } \rbrace \)  is a valid solution to the \((r-1)\)  rounds of the Sherali-Adams hierarchy (for CSPs). Then in order to finish Lemma REF  it suffices to give a randomized rounding algorithm that outputs 0-1 random variables \(\lbrace  X_v \rbrace _{v \in W}\)  such that \({\mathbb {E}}[X_v] = x_v = y_{pv}\)  for each \(v \in W\)  and \({\mathbb {E}}_{u, v \in W} {\mathbb {E}}[X_u X_v] = {\mathbb {E}}_{u, v} [x_{uv}] \pm \varepsilon _r = {\mathbb {E}}_{u, v}[y_{puv}] \pm \varepsilon _r\) . At this point, Theorem 4.6 of Raghavendra and Tan [1]} shows that such rounding a exists. For sake of completeness, we reproduce their proof here.
